Memoir on the General Theory of Surfaces And

CONTENTS Introduction. 79 1. Fundamental equations for a surface referred to its asymptotic curves. 85 2. Reciprocal congruences and the relation R. 86 3. The developables of the congruences T and r". 88 4. The focal points of the lines I and I'. 90 5. The directrix congruences. 91 6. Some general properties of reciprocal congruences. 93 7. The osculating quadric, and its connection-with reciprocal congruences. The canonical quadric. 96 8. Congruences conjugate and harmonic to a surface. 99 9. Congruences whose T-curves or r'-curves are indeterminate. 103 10. Planar nets with equal invariants. 106 11. The perspectives of the asymptotic curves on the tangent plane, and their four-point and osculating conies. 108 12. The canonical congruences of the first and second kinds. Characterization and generalization of scroll directrix congruences. Ill 13. Darboux's canonical expansion in non-homogeneous coordinates. 115 14. The completed characterization of the canonical tetrahedron. 122 15. The congruence of pseudo-normals. Í25 16. General theorems on conjugate nets. 130 17. Pseudo-geodesies, union curves, and generalized surfaces of Voss. 136 18. Dual considerations; curves of Darboux and Segre; axis and ray congruences. 140 19. Projective generalization of metric theorems. 144 20. Fundamental equations of the theory of surfaces. 148